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Branched coverings. (English) Zbl 0644.57001

J. W. Alexander [Bull. Amer. Math. Soc. 26, 370-372 (1920)] proved that each closed orientable PL n-manifold can be obtained as a branched covering over the n-sphere \(S^ n\). The author generalizes Alexander’s classical proof and shows that the natural setting for the result and proof is that of branched coverings between relative geometric cycles. The author shows that every geometric n-cycle is a branched covering over \(S^ n\). The author also proves several Hurwitz-like theorems on the existence and representation of branched coverings between relative geometric cycles. Many of the results of the paper exist in the folklore but are difficult to find in print.
Reviewer: J.W.Cannon

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
57Q99 PL-topology
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References:

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